Filter (mathematics): Difference between revisions
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In [[set theory]], a '''filter''' is a family of [[subset]]s of a given set which has properties generalising those of [[neighbourhood]] in [[topology]]. | In [[set theory]], a '''filter''' is a family of [[subset]]s of a given set which has properties generalising those of [[neighbourhood]] in [[topology]]. | ||
Revision as of 17:18, 27 November 2008
In set theory, a filter is a family of subsets of a given set which has properties generalising those of neighbourhood in topology.
Formally, a filter on a set X is a subset of the power set with the properties:
If G is a subset of X then the family
is a filter, the principal filter on G.
In a topological space , the neighbourhoods of a point x
form a filter, the neighbourhood filter of x.
Ultrafilters
An ultrafilter is a maximal filter: that is, a filter on a set which is not properly contained in any other filter on the set. Equivalently, it is a filter with the property that for any subset either or the complement .
The principal filter on a singleton set {x}, namely, all subsets of X containing x, is an ultrafilter.